Since the full equilibrium solution is too difficult to memorize, we’ll now present an approximation to the equilibrium solution that’s a little more reasonable to remember. Use this for situations when you have a CSI between 3 and 8, are far from the money, and no one has entered the pot before you.
Take the number of opponents to your left (so on the button, this would be 2).
Multiply this by your CSI (use fractions at your discretion).
At a full table, if there’s an ante bigger than one-fifth of the small blind, then subtract 5%.
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If the antes are smaller than one-fifth or if playing with 6 or fewer players, ignore the antes (i.e., don’t subtract 5%).This product is the Power Number (PN) you need in order to push all-in7.
Examples:
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Pushing from the button with a CSI of 4.5 needs a PN of 9.•
Pushing from 3 off the button with a CSI of 6 needs a PN of 30.•
Pushing from UTG at a 9-player table (with an ante) with a CSI of 5 needs a PN of 38 (8 opps x 5 CSI = 40.Subtract 5% for ante (2) = 38).
Now that you know what PN you need, the table on the facing page shows the PN of all the different starting hands.
Use the labels at the top and left of the chart; find one card along the left and the other across the top. Pairs are found along the diagonal of the chart. Suited cards are above and to the right of the diagonal. Offsuit cards are below and to the left of the diagonal. K9o has a PN of 12, A5s is 31, and pocket 3s are 33. Hands marked with a ‘+’
sign can be pushed from all positions with a CSI of 8 or lower. Hands without an entry are too weak to be pushed unless your CSI is less than 3.
Again, this doesn’t reflect the correct strengths of all hands in all situations; it’s an average strength over many different situations. There may be times where it’s preferable to push with a hand that has a lower PN than another (for example, the A5s in preference to AJo when pushing CSI 7 from 7 off the button).
When Your Opponents Don’t Play Optimally These equilibrium solutions assume that your opponents will be calling with the equilibrium calling range.
Those calling ranges are specified in Appendix 2.
What if you think that your opponents might have calling ranges that are tighter or looser than optimal? Like we said before, the good news is that you make more money either way. If you want to maximize your profits against suboptimal opponents, you should push more often against tighter opponents and push less often against loose ones. The tendencies of the blinds are the most important, since those will be the players who will most frequently call you.
Let’s look at a blind-stealing situation of raising all in on the button with a CSI of 5. The equilibrium solution—
again, far from the money—is to push with 22+, A2+, K9o+, K4s+, QTo+, Q8s+, JTo, J7s+, T7s+, 97s+, 86s+, 76s, 65s (35%). It assumes the SB will call with 22+, A2+, KTo+, K8s+, QJo, QTs+, JTs (26%) and the BB with 22+, A2+, K9o+, K6s+, QTo+, Q9s+, J9s+, T9s (30%). Players don’t often follow the equilibrium solution and call less frequently.
Looking at a large database of online hand histories, it appears that in this situation you’re much more likely to be called only about 16% from the SB and 17% from the BB. If this is true, you can profitably push 94% of hands! Naturally, if your opponents observe you pushing all the time, they’ll
start calling you more often. But until that point, they have no idea you’re pushing with such weak hands, unless someone calls and forces you to show. If you show down a weak hand and manage to win, you’ll have lost much of your fold equity. This should encourage you to push less frequently than the numbers suggest.
Now look at the other extreme example of pushing UTG at a 9-player table with a CSI of 7. The equilibrium solution (with antes) is to push with 66+, AQo+, ATs+, KTs+, QTs+, JTs (8.9%). It assumes that both blinds will call with 99+, AQ+ (5.1%) and everyone else with only 99+, AKo, AQs+ (4.2%). It always seems to me that my opponents aren’t capable of laying down AJo or ATs in this situation. Let’s assume that your opponents are a little looser so that everyone calls with 77+, AJ+, ATs (7.5%).
Against looser opponents it’s best to push only with 88+, AQo+, AJs+ (5.9%).
You should push more frequently if:
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Your remaining opponents are tighter than equilibrium (especially the blinds)—most applicable when stealing from the SB or button with a CSI above 4.•
You’re playing in a live tournament.•
You have a tight image.•
It has been awhile since you pushed all-in pre-flop.•
You’ve shown down strong hands after pushing.You should push less frequently if:
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Your remaining opponents are looser than equilibrium (especially the blinds)—most applicable when pushing into multiple opponents when you have a CSI above 6.•
You’re playing in an online tournament.•
You have a loose image.•
You have pushed all-in pre-flop several times in the last few hands.•
You’ve shown down some weak hands after pushing.In this chapter, the authors present many pertinent equations and mathematical formulas. However, I don’t necessarily suggest you learn them all by heart! Of course, in no-limit Texas hold ’em I believe that the mathematical aspect is very important, but also that it can have some limitations, especially when one starts accounting for less tangible factors, such as the skill levels of the players around the table, their respective positions, hand history, fold and fear equity, table dynamics, stack sizes, the stage of the tournament, the ambitions and objectives of your opponents, and on and on.
The charts you’ll find in the book, however, are very useful if you thoroughly familiarize yourself with them, as opposed to memorizing them, in order to analyze the relative strength of your hand as well as your opponents’ hand ranges. That said, I’m a strong believer that there’s no substitute for real-life hands-on experience. The more hands you play, the more capable and efficient you’ll become at evaluating hand ranges; this process almost becomes intuitive over time, with the experience you’ll accumulate. In short, the mathematical theory is essential and fascinating, but it’s mostly by playing the game that you develop your analytical skills.
Equilibrium play, as Lee and his cohorts describe it, refers to the optimal decision when one accounts for the greatest number of parameters that can be evaluated. However, if you watch the TV broadcast of any major poker tournament, you soon realize that players make sub-optimal decisions more often than not. What, then, becomes an interesting factor is the way you might be able to exploit and capitalize on your opponents’ suboptimal plays.
We all know that some players can never let go of pocket kings or aces. Against such profiles, you can capitalize, as long as you know how to enter pots against them in a relatively cheap and opportune manner—with suited connectors, for instance. If you connect with the flop, you’ll surely be able to stack off opponents who won’t lay down their premium hand, regardless of the texture of the board.
In my opinion, the concept of equilibrium play is particularly relevant in two precise situations: the pre-flop game; and endgame tournament play, when blinds are high relative to stack sizes. In both situations, I believe there’s a correct decision to make, dictated by mathematics. In all other situations, such as flop, turn, and river play, your main goal must be to exploit your opponents’ sub-optimal decisions.
A crucial factor to determine, based on your analysis of the table dynamics, is your opponents’ calling range, for their tournament life, that is. In my experience, most players are usually too tight in this category; their decision to lay down a big hand when they’re faced with calling a bet for all their
chips is often sub-optimal. Equilibrium plays, as the authors demonstrate, require players to call all-in more frequently than they actually do. Therefore, you can often make them fold a superior hand when the situation is right and you’re not afraid to commit all your chips.
Once again, be aware of the intangible factors that can generate sub-optimal decisions. For instance, I sometimes call an all-in bet with less than the right pots odds if I believe my opponent is a real threat and that the moment in the competition is right for me to take a shot at eliminating him or her right there. Conversely, I sometimes fold a hand against that same opponent, against pot odds, because I believe the timing isn’t right and I don’t want to give him (or her) an opportunity to double up against me.
Another parameter to which I pay close attention is the seating order, especially the seat that my dangerous opponents occupy relative to mine. Depending on the villain’s stack size and table position, I might adjust my decisions, sometimes going against what pure mathematics and pot odds dictate.
In summary, even though it’s fundamental to understand the mathematical notions underlying the game, you’ll be able to implement them only through intensive playing and the acquisition of the necessary experience to effectively analyze table dynamics, tendencies, and hand ranges.
Ultimately, you’ll learn to capitalize on sub-optimal decisions at the table.